Q3) a) Prove that the zero transformation \( T: V \rightarrow W \) is a linear transformation. b) Prove that the identity transformation \( T: V \rightarrow W \) is a linear transformation.

a) To prove that the zero transformation \( T: V \rightarrow W \) is a linear transformation, we need to show that it satisfies the two properties of linearity: 1. Additivity: For all \( u, v \in V \), \( T(u + v) = T(u) + T(v) \). 2. Homogeneity: For all \( u \in V \) and all scalars \( c \), \( T(cu) = cT(u) \). Let's prove each property: 1. Additivity: Since \( T \) is the zero transformation, \( T(u) = 0 \) and \( T(v) = 0 \) for all \( u, v \in V \). Therefore, \( T(u + v) = 0 \) (because \( u + v \) is also in \( V \)). On the other hand, \( T(u) + T(v) = 0 + 0 = 0 \). Thus, \( T(u + v) = T(u) + T(v) \). 2. Homogeneity: For any scalar \( c \) and \( u \in V \), we have \( T(cu) = 0 \) (because \( cu \) is also in \( V \)). On the other hand, \( cT(u) = c \cdot 0 = 0 \). Thus, \( T(cu) = cT(u) \). Since \( T \) satisfies both additivity and homogeneity, it is a linear transformation. b) To prove that the identity transformation \( T: V \rightarrow W \) is a linear transformation, we also need to show that it satisfies the two properties of linearity: 1. Additivity: For all \( u, v \in V \), \( T(u + v) = T(u) + T(v) \). 2. Homogeneity: For all \( u \in V \) and all scalars \( c \), \( T(cu) = cT(u) \). Let's prove each property: 1. Additivity: Since \( T \) is the identity transformation, \( T(u) = u \) and \( T(v) = v \) for all \( u, v \in V \). Therefore, \( T(u + v) = u + v \) (because \( u + v \) is also in \( V \)). On the other hand, \( T(u) + T(v) = u + v \). Thus, \( T(u + v) = T(u) + T(v) \). 2. Homogeneity: For any scalar \( c \) and \( u \in V \), we have \( T(cu) = cu \) (because \( cu \) is also in \( V \)). On the other hand, \( cT(u) = c \cdot u = cu \). Thus, \( T(cu) = cT(u) \). Since \( T \) satisfies both additivity and homogeneity, it is a linear transformation.